(B) Rolle's theorem requires the function $f(x)$ to be defined on a closed interval $[a, b]$, continuous on $[a, b]$, and differentiable on $(a, b)$.H

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Rolle's theorem is not applicable to $f$ in $D$

(B) Rolle's theorem requires the function $f(x)$ to be defined on a closed interval $[a, b]$, continuous on $[a, b]$, and differentiable on $(a, b)$.Here, the domain $D = [0, 1] \cup [2, 4]$ is not a single closed interval. It is a union of two disjoint closed intervals.For Rolle's theorem to be applicable, the domain must be a single connected closed interval $[a, b]$.Since the domain $D$ is disconnected, Rolle's theorem is not applicable to $f$ in $D$.

Class 12 Mathematics Continuity and Differentiability Rolle’s theorem, Lagrange's mean value theorem

Easy

Let $f: D \rightarrow R$ where $D=[0,1] \cup [2,4]$ be defined by $f(x)=\begin{cases} x, & \text{if } x \in [0,1] \\ 4-x, & \text{if } x \in [2,4] \end{cases}$. Then,

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A
Rolle's theorem is applicable to $f$ in $D$
B
Rolle's theorem is not applicable to $f$ in $D$
C
there exists $\xi \in D$ for which $f^{\prime}(\xi)=0$ but Rolle's theorem is not applicable
D
$f$ is not continuous in $D$

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Rolle’s theorem, Lagrange's mean value theorem

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In which of the following functions is Rolle's theorem applicable?

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Let $S$ be the set of all functions $f:[0,1] \rightarrow \mathbb{R}$ which are continuous on $[0,1]$ and differentiable on $(0,1)$. Then for every $f \in S$,there exists a $c \in (0,1)$,depending on $f$,such that:

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If $f(x) = \cos x$ for $0 \le x \le \frac{\pi}{2}$,then the real number $c$ of the Mean Value Theorem is:

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If a polynomial equation $a_nx^n + a_{n-1}x^{n-1} + \dots + a_2x^2 + a_1x + a_0 = 0$,where $n$ is a positive integer,has two distinct roots $\alpha$ and $\beta$,then how many roots does the equation $na_nx^{n-1} + (n - 1)a_{n-1}x^{n-2} + \dots + a_1 = 0$ have in the interval $(\alpha, \beta)$?

Medium

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If $2a + 3b + 6c = 0$,then at least one root of the equation $ax^2 + bx + c = 0$ lies in the interval:

DifficultAIEEE 2002

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Let $f: D \rightarrow R$ where $D=[0,1] \cup [2,4]$ be defined by $f(x)=\begin{cases} x, & \text{if } x \in [0,1] \\ 4-x, & \text{if } x \in [2,4] \end{cases}$. Then,

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In which of the following functions is Rolle's theorem applicable?

Let $S$ be the set of all functions $f:[0,1] \rightarrow \mathbb{R}$ which are continuous on $[0,1]$ and differentiable on $(0,1)$. Then for every $f \in S$,there exists a $c \in (0,1)$,depending on $f$,such that:

If $f(x) = \cos x$ for $0 \le x \le \frac{\pi}{2}$,then the real number $c$ of the Mean Value Theorem is:

If a polynomial equation $a_nx^n + a_{n-1}x^{n-1} + \dots + a_2x^2 + a_1x + a_0 = 0$,where $n$ is a positive integer,has two distinct roots $\alpha$ and $\beta$,then how many roots does the equation $na_nx^{n-1} + (n - 1)a_{n-1}x^{n-2} + \dots + a_1 = 0$ have in the interval $(\alpha, \beta)$?

If $2a + 3b + 6c = 0$,then at least one root of the equation $ax^2 + bx + c = 0$ lies in the interval:

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